Let \(\lbrace  M_i\rbrace \) 
be a defining sequence
for \(K\) . There exists an integer \(N\)  such that the diameter of each component of \(M_N\) 
is less than \(\varepsilon \) .
Enumerate all components
of \(M_N\) 
by \(M_N^{(1)},\ldots , M_N^{(s)}\) .
We may assume that for every \(i\) 
the set \(K\cap M_N^{(i)}\) 
is non-empty, hence is
a Cantor set.
Let \(K_i \subset K\cap M_N^{(i)}\) 
be a Cantor set which
is tame in \(\mathbb {R}^n\) 
(it can be constructed with the help
of Statement REF ),
and let \(B_i \subset \operatorname{Int}M_N^{(i)}\) 
be a tame Cantor set
all of whose projections onto hyperplanes are
\((n-1)\) -dimensional [1]}.
By Statement REF 
there exists
an isotopy \(\lbrace  h_t \rbrace  : \mathbb {R}^n\cong \mathbb {R}^n\) 
such that
\(h_t |_{\mathbb {R}^n - \cup _{i=1}^s M_N^{(i)} } = \operatorname{id}\)  for each \(t\in I\) ,
and \(h_1(K_i) = B_i\)  for each \(i=1,\ldots , s\) .
This is the desired map.
